A particle of mass m is executing oscillations about the origin on the x-axis. Its potential energy is U(x) = k[x]3, where k is a positive constant. If the amplitude of oscillation is a, then its time period T is

U = k|x|3  ⇒ F = -dUdx = -3k|x|2----------------(i)Also, for SHM x = a sin ωt and d2xdt2+ω2x= 0⇒acceleration = d2xdt2 = -ω2x  ⇒ F = ma= md2xdt2 = -mω2x----(ii)From equation (i) and (ii) we get ω = 3kxm⇒T = 2πω = 2πm3kx = 2πm3k(a sin ωt) ⇒ T ∝ 1a